Problem - 3866
(Apollonius’ Theorem) Let $AD$ be one median of $\triangle{ABC}$ where point $D$ lies on side $BC$. Show that the following relation holds:
$$AB^2 +AC^2 = 2\times(AD^2 +BD^2)$$
One way to prove this formula is to use the Steward Theorem (see # 2629).
$$AB^2\cdot CD + AC^2\cdot BD = BC \times (AD^2 + BD\cdot CD) $$
Setting $BC=2BD=2CD$ to the above relation immediately yields the result.